3H production via neutron-neutron-deuteron recombination A. Deltuva and A. C. Fonseca Centro de F´ısica Nuclear da Universidade de Lisboa, P-1649-003 Lisboa, Portugal (Received January 10, 2013) We study the recombination of two neutrons and deuteron into neutron and 3H using realistic nucleon-nucleon potential models. Exact Alt, Grassberger, and Sandhas equations for the four- nucleontransitionoperatorsaresolvedinthemomentum-spaceframeworkusingthecomplex-energy method with special integration weights. We find that at astrophysical or laboratory neutron den- sitiestheproductionof3Hviatheneutron-neutron-deuteronrecombination ismuchslower ascom- pared to the radiative neutron-deuteron capture. We also calculate neutron-3H elastic and total cross sections. PACSnumbers: 21.45.-v,25.10.+s,21.30.-x 3 1 0 I. INTRODUCTION the three-cluster breakup n + 3H n + n + d. Al- 2 though breakup reactions are usual→ly measured in nu- n Thenonrelativisticquantummechanicssolutionofthe clear physics, the recombination has the advantage that a its rate is finite at threshold where the breakup cross four-nucleon scattering problem has, in the past five J section vanishes due to phase-space factors. Further- years,reachedalevelofsophisticationandnumericalac- 9 more, n + n + d n + 3H is the only hadronic re- curacy that makes it a natural theoretical laboratory to → study nucleon-nucleon (NN) force models with the same combination reaction in the four-nucleon system that ] h confidence as one has used the three-nucleon system in at threshold is not suppressed by the Coulomb barrier t (like n + p + d p + 3H) or Pauli repulsion (like l- tmhaerpkafsotr. Tn-h3iHs haansdbepe-n3Hdeemeloanstsitcrastceadttienriangrecoebnsterbveanbclhes- n + n + n + p →n + 3H). It can take place in any c → environment with neutrons and deuterons and, with re- u [1], where three different theoretical frameworks have spect to the tritium synthesis, it may be competitive to n been compared, namely, the hyperspherical harmonics [ (HH) expansion method [2, 3], the Faddeev-Yakubovsky the electromagnetic capture reaction n+d γ +3H. → 1 (FY) equations [4] for the wave function components in Tnh+uns,+onde many+ri3sHe trheecoqmubesintiaotnioantwwohualdt cdoonmdiintiaotnesotvheer v coordinate space [5, 6], and the Alt, Grassberger and the n+d →γ+3Hradiativecaptureandtowhatextent 5 Sandhas(AGS)equations[7]fortransitionmatricesthat → 0 were solved in the momentum space [8, 9]. All methods it is relevant for astrophysical processes. 9 include not only the hadronic NN interaction, but also Inaddition,wealsopresentresultsforthen+3Helastic 1 the Coulomb repulsion between protons. While the first scattering and study the energy dependence of the total 1. two methods have the advantage of being able to deal n+3H cross section. 0 with charged-particle reactions at very low energies and 3 include static three-nucleon forces (3NF), the third one 1 is the only method so far to make predictions for multi- : v channelreactionssuchasd+d d+d, d+d n+3He, II. 4N SCATTERING EQUATIONS Xi d+d p+3H, and p+3H →n+3He (and→the corre- → → sponding inverse reactions) [10, 11]. r We use the time-reversal symmetry to relate the nnd a In a previous publication [12] a major step was taken recombination amplitude to the three-cluster breakup inextendingtheAGScalculationsabovethree-andfour- 3 amplitude of the initial n- H state, i.e., cluster breakup thresholds. Owing to the complicated analyticstructureofthefour-bodykernelabovebreakup thresholdthecalculationswereperformedusingthecom- Φ1 T13 Φ3 = Φ3 T31 Φ1 . (1) h | | i h | | i plex energy method [13, 14] whose accuracy and practi- cal applicability was greatly improved by a special in- Here Φ1 is the n-3H channel state and Φ3 is the nnd | i | i tegration method [12]. This allowed us to achieve fully channel state. The advantage is that the three-cluster converged results for n-3H elastic scattering with realis- breakup amplitude Φ3 T31 Φ1 is more directly related h | | i tic NN interactions. We note that the FY calculations to the AGS transition operators calculated in our βα ofn-3H elasticscatteringhavebeenrecentlyextendedas previous works [8, 12]. Since we uUse the isospin formal- welltoenergiesabovethefour-nucleonbreakupthreshold ismwhere the nucleonsare treatedas identicalfermions, [15],however,usingasemi-realisticNNpotentiallimited thereareonlytwodistincttwo-clusterpartitions,namely, to S-waves. β,α = 1 corresponds to the 3 + 1 partition (12,3)4 In the presentwork we extend the method of Ref. [12] whereasβ =2correspondstothe2+2partition(12)(34). to calculate the neutron-neutron-deuteron (nnd) recom- Forthe initial n-3Hstateweneedonly β1, i.e.,wesolve bination into n+3H and its time-reverse reaction, i.e., the AGS equations for the four-nucleonUtransition oper- 2 ators the total angular momentum of the (123) subsystem, s y and s are the spins of nucleons 3 and 4, S , S , and S −1 z x y z 11 = (G0tG0) P34 P34U1G0tG0 11 are the channel spins of two-, three-, and four-particle U − − U +U2G0tG0 21, (2a) systems, and is the total angular momentum with −1 U the projectionJ . We include a large number of 21 =(G0tG0) (1 P34)+(1 P34)U1G0tG0 11. M U − − U(2b) four-nucleon partial waves, lx,ly,lz,jx,jy,Jy ≤ 4 and 5, such that the results are well converged. The J ≤ complex-energy method [13] with special integration The free resolvent with the complex energy parameter weights [12] is used to treat the singularities of the Z =E+iε and the free Hamiltonian H0 is AGS equations (2). To obtain accurate results for the G0 =(Z H0)−1 (3) breakup amplitude hΦ3|T31|Φ1i near the nnd threshold − we had to use 0.1 MeV ε 0.4 MeV that are smaller ≤ ≤ whereas the pair (12) transition matrix than 1.0 MeV ε 2.0 MeV used in the elastic ≤ ≤ scattering calculations of Ref. [12]. However, the need t=v+vG0t (4) forrelativelysmallεvaluescausednotechnicalproblems since the integration with special weights [12] provides is derived from the respective potential v. The 3+1 and very accurate treatment of the 3H pole whereas the 2+2 subsystem transition operators are obtained from quasi-singularities due to deuteron pole are located the integral equations in a very narrow region with very small weight, such that about 30 grid points for the discretization of each −1 Uα =PαG0 +PαtG0Uα. (5) momentum variable were sufficient. The basis states are antisymmetric under exchange of the two nucleons (12). In the 2+2 partition the basis III. RESULTS states have to be antisymmetric also under exchange of thetwonucleons(34). Thefullantisymmetryasrequired The nnd recombination rate K3 is defined such that forthefour-nucleonsystemisensuredbythepermutation thenumberofrecombinationeventspervolumeandtime oPp13erPa2t3orasnPdaPb2o=f nPu1c3lePo2n4s. a and b with P1 = P12P23 + is K3ρ2nρd with ρn (ρd) being the density of neutrons (deuterons). We calculate it as a function of the relative 3 The n- H elastic and inelastic reaction amplitudes at the availableenergyE =ǫ1+p21/2µ1 are obtainedinthe nnd kinetic energy E3 =E−ǫd, i.e., mlpim1Niitsbεetihn→egrt+ehla0et.nivuHecelnero-e3nHǫ1mmiassostm.heeTnh3tuHemegl,araostnuidncdsµcs1atta=tteer3inemngeNar/gm4y-,, K3 =g3π2((2µπα)y7µµα1)p31/2E32 Xms Z d3kyd3kz (8) plitude is calculated in Refs. [8, 12]. The amplitude for k2 k2 the nnd breakup is obtained by the antisymmetrization Φ3 T31 Φ1 2δ E3 y z . ×|h | | i| − 2µαy − 2µα! of the general three-cluster breakup amplitude [16], re- sulting Here ǫ = 2.2246 MeV is the deuteron bound state en- d − ergy,µ andµ arethe reducedmassesassociatedwith hΦ3|T31|Φ1i=√3hΦ3|[(1−P34)U1G0tG0U11 (6) the fouαry-nucleonαJacobi momenta ky and kz. For exam- +U2G0tG0U21]|φ1i. ple,inthe2+2configurationky istherelativemomentum of the two neutrons while k is the relative momentum z Here φ1 is the Faddeev componentof the n-3H channel between the center of mass (c.m.) of the two-neutron | i state Φ1 = (1+P1)φ1 ; ǫ1 and φ1 are obtained by subsystemandthedeuteron. Thenndstatecanberepre- | i | i | i solving the bound-state Faddeev equation sented in both 3+1 and 2+2 configurations equally well; µ µ =m2 /2. The sum in Eq. (8) runs over all initial αy α N φ1 =G0tP1 φ1 (7) andfinalspinprojectionsm thatarenotexplicitlyindi- | i | i s catedinournotationforthechannelstateswhileg3 =12 at ε +0. → takes care of the spin averaging in the initial nnd state. We solve the AGS equations (2) in the momentum- The integral in Eq. (8), up to a factor, determines also space partial-wave framework. The momentum thetotalcrosssectionσ3 forthe three-clusterbreakupof and angular momentum part of the basis states the initial n-3H state. Thus, the nnd recombinationrate are k k k [l ( l [(l S )j s ]S J s )S ] | x y z z { y x x x y y} y z z JMi can be related to σ3 as for the 3 + 1 configuration and kxkykz(lz (lxSx)jx[ly(sysz)Sy]jy Sz) for the 8πg1p21 2| +2. Her{e kx, ky, and kz are th}e fouJr-Mpairticle Jacobi K3 = g3(µαyµα)3/2E32 σ3 (9) momenta as given in Ref. [17], l , l , and l are the x y z 3 corresponding orbital angular momenta, jx and jy are where g1 = 4 is the number of n- H spin states. Below the total angular momenta of pairs (12) and (34), Jy is the four-nucleon breakup threshold σ3 can be obtained 3 300 INOY04 ) r b/s CD Bonn 2 Phillips m 200 Seagrave Battat ( Ω INOY04 ) d b CD Bonn σd/ 100 σ (t 1 E = 9 MeV n 0 0 60 120 180 Θ (deg) c.m. 0 0 5 10 15 20 E (MeV) FIG. 1. (Color online) Differential cross section for elastic n n-3H scattering at 9 MeV neutron energy as a function of c.m. scattering angle. Results obtained with INOY04 (solid curves) and CD Bonn (dashed-dotted curves) potentials are FIG. 2. (Color online) Total cross section for n-3H scatter- compared with theexperimental data from Ref. [22]. ingasafunction of theneutronlab energy. Resultsobtained with INOY04 (solid curves) and CD Bonn (dashed-dotted curves) potentials are compared with the experimental data from Refs. [23, 24]. via the optical theorem as a difference between the to- tal and elastic cross sections. The equation (9) cannot be used right at the nnd threshold where both E3 and σ3 vanish. For E3 0 the nnd recombination rate (8) and compare it to the data of Refs. [23, 24]. The three- → becomes cluster (four-cluster) breakup threshold corresponds to E = 8.35 (11.31) MeV. As already found in Refs. n K30 = 4π(2π)7µ1p1 Φ03 T31 Φ1 2, (10) [5, 8, 25], the total n-3H cross section around the low- g3 |h | | i| energy peak is underpredicted by the traditional two- Xms nucleon potentials while the low-momentumor chiralef- where for the channel state Φ0 the relative momenta fective field theory potentials may provide a better de- 3 | i k = k = 0. The most convenient representation for scription [26, 27]. Although with increasing energy the y z Φ0 is a single-component 2+2 state with l = l = predictions approach the experimental data, as already 3 y z | i S =j =0 and j =S = =1. mentioned [8], the elastic and total cross section data y y x z J We study the four-nucleonsystemusing realistichigh- maybe inconsistent. Inthe low-energyregimewherethe precision two-nucleon potentials, namely, the inside- inelastic cross section should vanish for En 8.35 MeV ≤ nonlocal outside-Yukawa (INOY04) potential by Do- andremainverysmallatEn =9MeV,thereisingeneral leschall [5, 18], the Argonne (AV18) potential [19], the a better agreement between predictions and experiment charge-dependent Bonn potential (CD Bonn) [20], and for the elastic differential cross section than for the to- its extension CD Bonn + ∆ [21] allowing for an exci- talcrosssectionwhichissignificantlyunderestimatedby tation of a nucleon to a ∆ isobar and thereby yielding theory. A solution to this discrepancy may require new effective three- and four-nucleon forces. Among these measurements in this energy regime. potentials only INOY04 nearly reproduces experimental In Fig. 3 we study the energy-dependence of the nnd baninddiCnDg eBneorngny+of3∆Hu(n8d.4e8rbMinedV)t,hweh3iHlenAuVc1le8u,sCbDyB0o.8n6n, rNecomisbtihneatAiovnograadteroi’ns nthuembstearn.dWaredsfhoormw oNnA2lyKI3NwOhYe0r4e A 0.48 and 0.20 MeV, respectively. predictions as it is the only used potential with correct First we study the n-3H reactions for existing experi- ǫ1 and p1 values. The results at E3 = 0 are obtained mental data. We concentrate on the energy regime rele- from Eq. (10) while at E3 >0 it was more convenientto vant for the nnd recombination, i.e., between the three- use Eq. (9) where σ3 was calculated using optical theo- and four-cluster breakup thresholds. In Fig. 1 we show rem. Thus, for E3 > ǫd our predictions in Fig. 3 esti- thedifferentialcrosssectionforn-3Helasticscatteringat mate the upper limit f|or|NA2K3 since they assume that En =9 MeV neutronenergycorrespondingto E3 =0.49 the four-cluster breakup cross section is much smaller MeV. The predictions agree well with the experimental than the three-cluster breakupcrosssection. In the rele- dataofRef.[22]andarequiteinsensitivetothe choiceof vantenergy regime 0 E3 ǫd the nnd recombination 3 ≤ ≤| | the potential. Results for n- H elastic scattering above rate increases with increasing energy E3 nearly linearly the four-clusterbreakupthresholdupto E =22.1MeV due to the increasing contributions of partial waveswith n are given in Ref. [12]. nonzero orbital angular momentum l . The threshold z 3 2 0 InFig.2weshowthetotalcrosssectionforn- Hscat- valuesN K referringtoallemployedpotentialsarecol- A 3 tering at neutron energies ranging from 0 to 22 MeV lected in Table I; they increase with 3H binding energy. 4 2) nation is entirely irrelevant as well as for the big-bang -ol nucleosynthesis where the estimated baryon density is m 1 3 × 10-5 even lower. On the other hand, the neutron density in -s core-collapse supernova or neutron stars may be higher 6m than ρc by several orders of magnitude but the absence c 2 × 10-5 of deunterons renders n + d and n + n + d reactions ( 3 irrelevant. However, based on our results one may K 2N A 1 × 10-5 conjecture that at such high densities the three-cluster recombination of two neutrons and a heavier nucleus A, 0 1 2 3 4 i.e., n+n+A n+(An) might be as important as E3 (MeV) the correspondin→g radiative capture n+A γ+(An). For example, the above reactions with A be→ing 20Ne are relevant for the neon-burning process. FIG. 3. (Color online) nnd recombination rate K3 as a IV. SUMMARY function of relative kinetic nnd energy E3. Predictions are obtained using theINOY04potential. We havesolvedthe four-nucleonAGSequationsinthe energyregimeabovethethree-clusterthresholdandstud- AV18 |ǫ1|7(.M62eV) NA2K130.3(c1m×6s1−01−m5ol−2) ined+t3hHe.raTtheefoorbttahienerdecroemsublitnsasthioonwrethacattiotnhenn+ndnr+ecdom→- CD Bonn 8.00 1.41×10−5 binationisnotcompetitivewiththeradiativendcapture for the production of tritium at neutron densities avail- CD Bonn + ∆ 8.28 1.47×10−5 able in laboratory induced fusion or astrophysical pro- INOY04 8.49 1.52×10−5 cesses. Thus, one may conjecture with a confidence that other nucleon-nucleon-deuteron recombination reactions TABLE I. nnd recombination rate at threshold calculated (forexample,p+p+d p+3Hethatcouldcontributeto withdifferenttwo-nucleonpotentials. Thevaluesfor3Hbind- thehydrogenburningp→rocessinstars),beinginaddition ing energy are listed as well. suppressedby the Coulombrepulsion, are inferior to the respective nucleon-deuteron radiative capture reactions at realistic densities, and that four-nucleon recombina- Finally we compare the relative importance of the tion reactions are even far less relevant. nnd recombination and nd radiative capture. For the In addition, we presented results for the n-3H elastic latterthenumberofevents,i.e.,the numberofproduced differential cross section at En = 9 MeV and the n-3H 3H nuclei per volume and time is K2ρnρd with K2 totalcrosssectionup toEn =22MeV.While the elastic being the nd capture rate. The threshold value for it cross section agrees fairly well with the data, there is a given in Ref. [28] is N K0 = 66.2cm3s−1mol−1. The disagreementforthe totalcrosssection,especially in the A 2 critical density of neutrons at which both processes low-energyregime. 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